math.log10 on complex, real part

Average Error: 32.1 → 17.6

Time: 6.5s

Precision: 64

Math

\[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]

↓

\[\begin{array}{l} \mathbf{if}\;re \le -5.543624575485825300470836787098407507184 \cdot 10^{99}:\\ \;\;\;\;\log \left({\left(\frac{-1}{re}\right)}^{\left(-\sqrt{\frac{1}{\log 10}}\right)}\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \mathbf{elif}\;re \le -1.005507393154052661497216237161906591061 \cdot 10^{-263}:\\ \;\;\;\;\log \left(\sqrt[3]{{\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}}\right) \cdot \frac{2}{\sqrt{\log 10}} + \frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt[3]{{\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}}\right)\\ \mathbf{elif}\;re \le 2.695186912474015462769700121112449872029 \cdot 10^{-166}:\\ \;\;\;\;\log \left({im}^{\left(\sqrt{\frac{1}{\log 10}}\right)}\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \mathbf{elif}\;re \le 4.969148025702489596813788314716096306133 \cdot 10^{122}:\\ \;\;\;\;\log \left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \mathbf{else}:\\ \;\;\;\;\log \left({re}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \end{array}\]

C

double f(double re, double im) {
        double r49706 = re;
        double r49707 = r49706 * r49706;
        double r49708 = im;
        double r49709 = r49708 * r49708;
        double r49710 = r49707 + r49709;
        double r49711 = sqrt(r49710);
        double r49712 = log(r49711);
        double r49713 = 10.0;
        double r49714 = log(r49713);
        double r49715 = r49712 / r49714;
        return r49715;
}

↓

double f(double re, double im) {
        double r49716 = re;
        double r49717 = -5.543624575485825e+99;
        bool r49718 = r49716 <= r49717;
        double r49719 = -1.0;
        double r49720 = r49719 / r49716;
        double r49721 = 1.0;
        double r49722 = 10.0;
        double r49723 = log(r49722);
        double r49724 = r49721 / r49723;
        double r49725 = sqrt(r49724);
        double r49726 = -r49725;
        double r49727 = pow(r49720, r49726);
        double r49728 = log(r49727);
        double r49729 = sqrt(r49723);
        double r49730 = r49721 / r49729;
        double r49731 = r49728 * r49730;
        double r49732 = -1.0055073931540527e-263;
        bool r49733 = r49716 <= r49732;
        double r49734 = r49716 * r49716;
        double r49735 = im;
        double r49736 = r49735 * r49735;
        double r49737 = r49734 + r49736;
        double r49738 = sqrt(r49737);
        double r49739 = pow(r49738, r49730);
        double r49740 = cbrt(r49739);
        double r49741 = log(r49740);
        double r49742 = 2.0;
        double r49743 = r49742 / r49729;
        double r49744 = r49741 * r49743;
        double r49745 = r49730 * r49741;
        double r49746 = r49744 + r49745;
        double r49747 = 2.6951869124740155e-166;
        bool r49748 = r49716 <= r49747;
        double r49749 = pow(r49735, r49725);
        double r49750 = log(r49749);
        double r49751 = r49750 * r49730;
        double r49752 = 4.9691480257024896e+122;
        bool r49753 = r49716 <= r49752;
        double r49754 = log(r49739);
        double r49755 = r49754 * r49730;
        double r49756 = pow(r49716, r49730);
        double r49757 = log(r49756);
        double r49758 = r49757 * r49730;
        double r49759 = r49753 ? r49755 : r49758;
        double r49760 = r49748 ? r49751 : r49759;
        double r49761 = r49733 ? r49746 : r49760;
        double r49762 = r49718 ? r49731 : r49761;
        return r49762;
}

Error

Bits error versus re

Bits error versus im

Derivation

1. Split input into 5 regimes

2. if re < -5.543624575485825e+99

   1. Initial program 50.9

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 50.9

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\color{blue}{\sqrt{\log 10} \cdot \sqrt{\log 10}}}\]

   4. Applied pow1 50.9

      \[\leadsto \frac{\log \color{blue}{\left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^{1}\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

   5. Applied log-pow 50.9

      \[\leadsto \frac{\color{blue}{1 \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

   6. Applied times-frac 50.9

      \[\leadsto \color{blue}{\frac{1}{\sqrt{\log 10}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\sqrt{\log 10}}}\]
   7. Using strategy rm

   8. Applied add-log-exp 50.9

      \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \color{blue}{\log \left(e^{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\sqrt{\log 10}}}\right)}\]

   9. Simplified 50.8

      \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \log \color{blue}{\left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)}\]
   10. Using strategy rm

   11. Applied *-commutative 50.8

       \[\leadsto \color{blue}{\log \left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right) \cdot \frac{1}{\sqrt{\log 10}}}\]

   12. Taylor expanded around -inf 9.1

       \[\leadsto \log \color{blue}{\left(e^{-1 \cdot \left(\log \left(\frac{-1}{re}\right) \cdot \sqrt{\frac{1}{\log 10}}\right)}\right)} \cdot \frac{1}{\sqrt{\log 10}}\]

   13. Simplified 9.0

       \[\leadsto \log \color{blue}{\left({\left(\frac{-1}{re}\right)}^{\left(-\sqrt{\frac{1}{\log 10}}\right)}\right)} \cdot \frac{1}{\sqrt{\log 10}}\]

   if -5.543624575485825e+99 < re < -1.0055073931540527e-263

   1. Initial program 20.0

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 20.0

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\color{blue}{\sqrt{\log 10} \cdot \sqrt{\log 10}}}\]

   4. Applied pow1 20.0

      \[\leadsto \frac{\log \color{blue}{\left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^{1}\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

   5. Applied log-pow 20.0

      \[\leadsto \frac{\color{blue}{1 \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

   6. Applied times-frac 20.0

      \[\leadsto \color{blue}{\frac{1}{\sqrt{\log 10}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\sqrt{\log 10}}}\]
   7. Using strategy rm

   8. Applied add-log-exp 20.0

      \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \color{blue}{\log \left(e^{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\sqrt{\log 10}}}\right)}\]

   9. Simplified 19.8

      \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \log \color{blue}{\left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)}\]
   10. Using strategy rm

   11. Applied add-cube-cbrt 19.8

       \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \log \color{blue}{\left(\left(\sqrt[3]{{\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}} \cdot \sqrt[3]{{\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}}\right) \cdot \sqrt[3]{{\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}}\right)}\]

   12. Applied log-prod 19.9

       \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \color{blue}{\left(\log \left(\sqrt[3]{{\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}} \cdot \sqrt[3]{{\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}}\right) + \log \left(\sqrt[3]{{\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}}\right)\right)}\]

   13. Applied distribute-lft-in 19.9

       \[\leadsto \color{blue}{\frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt[3]{{\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}} \cdot \sqrt[3]{{\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}}\right) + \frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt[3]{{\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}}\right)}\]

   14. Simplified 19.9

       \[\leadsto \color{blue}{\log \left(\sqrt[3]{{\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}}\right) \cdot \frac{2}{\sqrt{\log 10}}} + \frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt[3]{{\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}}\right)\]

   if -1.0055073931540527e-263 < re < 2.6951869124740155e-166

   1. Initial program 32.3

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 32.3

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\color{blue}{\sqrt{\log 10} \cdot \sqrt{\log 10}}}\]

   4. Applied pow1 32.3

      \[\leadsto \frac{\log \color{blue}{\left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^{1}\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

   5. Applied log-pow 32.3

      \[\leadsto \frac{\color{blue}{1 \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

   6. Applied times-frac 32.3

      \[\leadsto \color{blue}{\frac{1}{\sqrt{\log 10}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\sqrt{\log 10}}}\]
   7. Using strategy rm

   8. Applied add-log-exp 32.3

      \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \color{blue}{\log \left(e^{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\sqrt{\log 10}}}\right)}\]

   9. Simplified 32.1

      \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \log \color{blue}{\left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)}\]
   10. Using strategy rm

   11. Applied *-commutative 32.1

       \[\leadsto \color{blue}{\log \left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right) \cdot \frac{1}{\sqrt{\log 10}}}\]

   12. Taylor expanded around 0 34.5

       \[\leadsto \log \color{blue}{\left(e^{\log im \cdot \sqrt{\frac{1}{\log 10}}}\right)} \cdot \frac{1}{\sqrt{\log 10}}\]

   13. Simplified 34.5

       \[\leadsto \log \color{blue}{\left({im}^{\left(\sqrt{\frac{1}{\log 10}}\right)}\right)} \cdot \frac{1}{\sqrt{\log 10}}\]

   if 2.6951869124740155e-166 < re < 4.9691480257024896e+122

   1. Initial program 16.7

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 16.7

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\color{blue}{\sqrt{\log 10} \cdot \sqrt{\log 10}}}\]

   4. Applied pow1 16.7

      \[\leadsto \frac{\log \color{blue}{\left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^{1}\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

   5. Applied log-pow 16.7

      \[\leadsto \frac{\color{blue}{1 \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

   6. Applied times-frac 16.6

      \[\leadsto \color{blue}{\frac{1}{\sqrt{\log 10}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\sqrt{\log 10}}}\]
   7. Using strategy rm

   8. Applied add-log-exp 16.6

      \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \color{blue}{\log \left(e^{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\sqrt{\log 10}}}\right)}\]

   9. Simplified 16.4

      \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \log \color{blue}{\left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)}\]
   10. Using strategy rm

   11. Applied *-commutative 16.4

       \[\leadsto \color{blue}{\log \left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right) \cdot \frac{1}{\sqrt{\log 10}}}\]

   if 4.9691480257024896e+122 < re

   1. Initial program 56.3

      \[\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\log 10}\]
   2. Using strategy rm

   3. Applied add-sqr-sqrt 56.3

      \[\leadsto \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\color{blue}{\sqrt{\log 10} \cdot \sqrt{\log 10}}}\]

   4. Applied pow1 56.3

      \[\leadsto \frac{\log \color{blue}{\left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^{1}\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

   5. Applied log-pow 56.3

      \[\leadsto \frac{\color{blue}{1 \cdot \log \left(\sqrt{re \cdot re + im \cdot im}\right)}}{\sqrt{\log 10} \cdot \sqrt{\log 10}}\]

   6. Applied times-frac 56.3

      \[\leadsto \color{blue}{\frac{1}{\sqrt{\log 10}} \cdot \frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\sqrt{\log 10}}}\]
   7. Using strategy rm

   8. Applied add-log-exp 56.3

      \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \color{blue}{\log \left(e^{\frac{\log \left(\sqrt{re \cdot re + im \cdot im}\right)}{\sqrt{\log 10}}}\right)}\]

   9. Simplified 56.3

      \[\leadsto \frac{1}{\sqrt{\log 10}} \cdot \log \color{blue}{\left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right)}\]
   10. Using strategy rm

   11. Applied *-commutative 56.3

       \[\leadsto \color{blue}{\log \left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right) \cdot \frac{1}{\sqrt{\log 10}}}\]

   12. Taylor expanded around inf 8.1

       \[\leadsto \log \left({\color{blue}{re}}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right) \cdot \frac{1}{\sqrt{\log 10}}\]
3. Recombined 5 regimes into one program.

4. Final simplification 17.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;re \le -5.543624575485825300470836787098407507184 \cdot 10^{99}:\\ \;\;\;\;\log \left({\left(\frac{-1}{re}\right)}^{\left(-\sqrt{\frac{1}{\log 10}}\right)}\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \mathbf{elif}\;re \le -1.005507393154052661497216237161906591061 \cdot 10^{-263}:\\ \;\;\;\;\log \left(\sqrt[3]{{\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}}\right) \cdot \frac{2}{\sqrt{\log 10}} + \frac{1}{\sqrt{\log 10}} \cdot \log \left(\sqrt[3]{{\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}}\right)\\ \mathbf{elif}\;re \le 2.695186912474015462769700121112449872029 \cdot 10^{-166}:\\ \;\;\;\;\log \left({im}^{\left(\sqrt{\frac{1}{\log 10}}\right)}\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \mathbf{elif}\;re \le 4.969148025702489596813788314716096306133 \cdot 10^{122}:\\ \;\;\;\;\log \left({\left(\sqrt{re \cdot re + im \cdot im}\right)}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \mathbf{else}:\\ \;\;\;\;\log \left({re}^{\left(\frac{1}{\sqrt{\log 10}}\right)}\right) \cdot \frac{1}{\sqrt{\log 10}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019353 
(FPCore (re im)
  :name "math.log10 on complex, real part"
  :precision binary64
  (/ (log (sqrt (+ (* re re) (* im im)))) (log 10)))